Page 30 - Matrices theory and applications

1.3. Exercises
If B : E × F → K is bilinear, one can compare the matrices M and

M of B with respect to the bases β, γ and β ,γ .Denoting by P, Q the

change-of-basis matrices of β → β and γ → γ ,one has

j
ij
i
k,l
k,l
Therefore, m = B(e ,f )= p ki q lj B(e k ,f l )= p ki q lj m kl . 13
T
M = P MQ.

When F = E and γ = β, γ = β , the change of basis has the effect of

T
replacing M by M = P MP. In general, M is not similar to M, though

it is so if P is orthogonal. If M is symmetric, then M is too. This was
expected, since one expresses the symmetry of the underlying bilinear form
B.
If the characteristic of K is distinct from 2, there is an isomorphism
n
between Sym (K) and the set of quadratic forms on K . This isomorphism
n
is given by the formula
Q(e i + e j ) − Q(e i ) − Q(e j )= 2m ij .
In particular, Q(e i )= m ii .
1.3 Exercises
1. Let G be an IR-vector space. Verify that its complexification G CC is a
CC-vector space and that dim CC G CC =dim IR G.

2. Let M ∈ M n×m (K)and M ∈ M m×p (K) be given. Show that

rk(MM ) ≤ min{rk M, rk M }.

First show that rk(MM ) ≤ rk M, and then apply this result to the

transpose matrix.
3. Let K be a field and let A, B, C be matrices with entries in K,of
respective sizes n × m, m × p,and p × q.
(a) Show that rk A +rk B ≤ m +rk AB. It is sufficient to consider
the case where B is onto, by considering the restriction of A to
the range of B.
(b) Show that rk AB +rk BC ≤ rk B +rk ABC.One may use
p
the vector spaces K / ker B and R(B), and construct three
homomorphisms u, v, w,with v being onto.

∗
4. (a) Let n, n ,m,m ∈ IN and let K be a field. If B ∈ M n×m(K)and
C ∈ M n ×m (K), one defines a matrix B ⊗ C ∈ M nn ×mm (K),